The Ventilated Ocean

The Ventilated Ocean

PATRICK HAERTEL AND ALEXEY FEDOROV

Geology and Geophysics, Yale University, New Haven, Connecticut

(Manuscript received 22 September 2010, in final form 27 July 2011)

ABSTRACT

Adiabatic theories of ocean circulation and density structure have a long tradition, from the concept of the

ventilated thermocline to the notion that deep ocean ventilation is controlled by westerly winds over the

Southern Ocean. This study explores these ideas using a recently developed Lagrangian ocean model (LOM),

which simulates ocean motions by computing trajectories of water parcels. A unique feature of the LOM is its

capacity to model ocean circulations in the adiabatic limit, in which water parcels exactly conserve their

densities when they are not in contact with the ocean surface. The authors take advantage of this property of

the LOM and consider the circulation and stratification that develop in an ocean with a fully adiabatic interior

(with both isopycnal and diapycnal diffusivities set to zero). The ocean basin in the study mimics that of the

Atlantic Ocean and includes a circumpolar channel. The model is forced by zonal wind stress and a density

restoring at the surface.

Despite the idealized geometry, the relatively coarse model resolution, and the lack of atmospheric cou-

pling, the nondiffusive ocean maintains a density structure and meridional overturning that are broadly in line

with those observed in the Atlantic Ocean. These are generated by just a handful of key water pathways,

including shallow tropical cells described by ventilated thermocline theory; a deep overturning cell in which

sinking North Atlantic Deep Water eventually upwells in the Southern Ocean before returning northward as

Antarctic Intermediate Water; a Deacon cell that results from a topographically steered and corkscrewing

circumpolar current; and weakly overturning Antarctic Bottom Water, which is effectively ventilated only in

the Southern Hemisphere.

The main conclusion of this study is that the adiabatic limit for the ocean interior provides the leading-order

solution for ocean overturning and density structure, with tracer diffusion contributing first-order pertur-

bations. Comparing nondiffusive and diffusive experiments helps to quantify the changes in stratification and

circulation that result from adding a moderate amount of tracer diffusion in the ocean model, and these

include an increase in the amplitude of the deep meridional overturning cell of several Sverdrups, a 10%–20%

increase in Northern Hemispheric northward heat transport, a stronger stratification just below the main

thermocline, and a more realistic bottom overturning cell.

1. Introduction

For more than 70 years oceanographers have been

aware that vertical variations in density in the oceans

could result from adiabatic transports along isopycnal

surfaces of water masses formed at different latitudes

(Pedlosky 1996). Montgomery (1938) and Iselin (1939)

were the first to introduce such a concept of thermocline

ventilation. Welander (1959) proposed the first adiabatic

thermocline model based on ideal-fluid equations. The

adiabatic framework was further discussed by Veronis

(1969). Luyten Pedlosky and Stommel (1983) and Huang

(1988) formalized this idea into a rigorous adiabatic the-

ory for the density and flow structure of the upper ocean.

These ventilated thermocline theories, however, are in-

complete in that they require a background stratifica-

tion, which is typically defined at the eastern boundary of

the ocean basin. Traditionally, it has been thought that

deep ocean thermal structure—and hence the background

stratification needed for adiabatic theories—is determined

by a balance between vertical advection and vertical dif-

fusion of heat (e.g., Robinson and Stommel 1959). In fact,

numerous studies of ocean thermohaline circulation in

closed basins invoke this balance as a driving mechanism

for the deep ocean circulation (e.g., Bryan 1987; Huang

1999; Park and Bryan 2000; Fedorov et al. 2007).

The tension between adiabatic and diabatic ways of

thinking about ocean stratification and circulation is

Corresponding author address: Dr. Patrick T. Haertel, Geology

and Geophysics, Yale University, 210 Whitney Ave., New Haven,

CT 06510.

E-mail: [email protected]

JANUARY 2012 H A E R T E L A N D F E D O R O V 141

DOI: 10.1175/2011JPO4590.1

� 2012 American Meteorological Society

reflected in recent studies of ocean thermal structure

(Tziperman 1986; Samelson and Vallis 1997; Samelson

2004; Boccaletti et al. 2004; Fedorov et al. 2010), ocean

energetics (Wunsch and Ferrari 2004), responses to fresh-

water fluxes (e.g., Barreiro et al. 2008; Fedorov et al.

2004), and different observations of ocean vertical mixing

(Ledwell et al. 1993; Polzin et al. 1997). Moreover, the

work of Toggweiler and Samuels (1995, 1998) and sub-

sequent studies (e.g., Gnanadesikan 1999; Gnanadesikan

and Hallberg 2000; Bryden and Cunningham 2003) point

to the important role of Southern Ocean processes such

as wind-driven upwelling and eddy fluxes in maintaining

the global ocean density structure and deep ocean over-

turning. Most recently, these ideas have been explored in

global ocean models (Saenko 2007; Kuhlbrodt et al. 2007),

idealized eddy-resolving models of the Atlantic (Wolfe

and Cessi 2010), and zonally averaged ocean models

(Sevellec and Fedorov 2011).

While it is likely that ocean stratification and circulation

are determined by a combination of all aforementioned

processes, the question arises whether oceanic meridional

overturning and density structure can be maintained even

in the complete absence of interior tracer diffusion. The

major goal of our study is to address this important

question. We use a recently developed Lagrangian ocean

model (LOM) (Haertel and Randall 2002; Haertel et al.

2004, 2009; Van Roekel et al. 2009) to examine the effects

of completely removing interior tracer diffusion (both

diapycnal and isopycnal) on ocean meridional overturning

and stratification. We employ an idealized basin with

sloping boundaries that includes an extended periodic

channel, which captures the gross geometry of the Atlantic

Ocean and the Antarctic circumpolar channel.

Our study is in the spirit of Toggweiler and Samuels

(1995), and especially Toggweiler and Samuels (1998),

who considered ocean stratification and meridional over-

turning as tracer mixing approached zero. Their modeling

results were obtained within a traditional z-coordinate

model and were subject to an uncertain amount of spurious

numerical mixing (Griffies et al. 2000), as well as explicit

horizontal and low vertical diffusion. Here, we actually

conduct a simulation in the fully adiabatic limit, albeit for

a more idealized setting. We exploit a unique capability of

the LOM: the capacity to conduct runs with a tracer dif-

fusivity set to zero, with water parcels exactly conserving

their densities in the ocean interior. We also take advantage

of the fact that the LOM provides trajectories for every

water parcel in the ocean and examine these to determine

locations where water masses form, where sinking water

ultimately upwells, and how the zonally integrated merid-

ional overturning streamfunction is partitioned into cross-

hemispheric and Southern Hemispheric components.

Finally, we repeat our simulation with a moderate amount

of tracer diffusion so that we can precisely quantify its

contributions to stratification and circulation for our ide-

alized ocean.

The model that we use (the LOM) has several key

differences from traditional isopycnal models. First, the

LOM has no spurious numerical isopycnal diffusion

associated with advection, which means there is no nu-

merical degradation of tracer distributions and the mod-

eler has explicit control over the amount of isopycnal

mixing. This feature can be quite useful for simulating

both dynamical and biogeochemical tracer distributions,

and it is especially important for the long time scales

needed for adjustment of the deep ocean. Second, the

behavior at the smallest resolvable scales is also unique

in the LOM, with spontaneous generation of bolus

transports (see section 3e). In addition, unlike isopycnal

models, the LOM does not require diffusion of isopycnal

layer thicknesses for numerical stability. Finally, the

LOM’s convective parameterization allows for a convec-

tive redistribution of water mass in the vertical that does

not involve mixing. In other words, the convective scheme

can represent plumes that are completely nonentraining.

Because of these model properties we expect adiabatic

LOM simulations to show some key differences from

isopycnal simulations with no vertical mixing.

This paper is organized as follows. Section 2 outlines the

design of our modeling experiments. In section 3 we ex-

amine the circulation and density structure of an idealized

ocean in the zero-diffusivity (adiabatic) limit and then

consider how the presence of tracer diffusion alters strat-

ification, overturning, and key water pathways. Section 4 is

a summary and discussion. Details of the LOM and results

of several sensitivity tests are provided in the appendix.

2. Experimental design

We carry out a series of modeling experiments that

examine to what extent surface forcing and adiabatic

ocean circulations alone determine the gross stratification

and meridional overturning of an idealized ocean. The

experiments take advantage of the Lagrangian formula-

tion of our ocean model (discussed in the appendix), both

in terms of its ability to simulate circulations with zero

tracer diffusion and its capacity to track every mass ele-

ment in the ocean. This section describes the idealized

ocean, surface forcing, and spinup procedure for the sim-

ulations discussed in the remainder of the paper.

a. Idealized ocean

The basin geometry for the idealized ocean is illus-

trated in Fig. 1. The model domain extends from 708S to

708N, and there is a periodic channel that runs along the

southern boundary. The maximum depth of the ocean is

142 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42

4.9 km, and the channel is 2.1 km deep. In most of the

domain the ocean is about 608 longitude wide, but we

make the channel much wider (1778 longitude) to create

a large region where water can upwell along the southern

boundary, which makes the meridional overturning

streamfunction more like that of the World Ocean (see

below). Because we use a Mercator projection, the actual

length of our channel is roughly the same as that of the

Antarctic ‘‘Circumpolar Channel.’’ The geometry of the

setting resembles that of Wolfe and Cessi (2009, 2010),

with several differences: our basin has sloping bound-

aries, the circumpolar channel is significantly longer, and

the depth of basin is almost 5 km (twice that of the pre-

vious studies).

For our featured experiments, we represent this ocean

as a collection of rather large water parcels with a 38 radius

in latitude and longitude and a vertical thickness of about

78 m. Sensitivity tests show that our key results are rela-

tively insensitive to a factor of 2 changes in the horizontal

and vertical resolutions of the model (see the appendix).

We use a time step of 12 000 s, which is facilitated by using

a form of gravity wave retardation (Jensen 1996) to slow

external gravity waves by roughly a factor of 10 (see ap-

pendix). During the spinup simulation and for supple-

mental simulations mass elements in the middle and

bottom of the ocean are moved in groups of two and four,

respectively, for computational efficiency. This approach

is equivalent to using a lower vertical resolution in these

regions: it was first used by Haertel et al. (2009).

No horizontal mixing of tracers is used, but we have

found that even at low resolution the model generates

a bolus transport of isopycnal layer thickness similar to

that which a Gent and McWilliams (1990, hereafter GM)

parameterization would produce. This important prop-

erty of the LOM is discussed further in section 3e.

b. Surface forcing

The surface forcings that we apply to the idealized

ocean are depicted in Fig. 2. The zonal wind stress (Fig. 2a)

is an analytical approximation of the observed wind stress

over the Atlantic. However, despite the fact that observed

westerlies are stronger in the Southern Hemisphere than

in the Northern Hemisphere, we use a symmetric wind

forcing so that the primary asymmetry about the equator

in our model comes from including the periodic channel.

Surface density is restored to a piecewise linear function

(Fig. 2b, black line) in most of the model domain. If den-

sity is converted to temperature, the surface restoring

equates to a 50 W m22 8C21 heat flux, or a piston velocity

(Griffies et al. 2009) of about 1 m day21. The minimum

and maximum density for the restoring function are cho-

sen so that simulated surface density fields span the same

range as the observed surface density field in the vicinity of

308W (note that the simulated equatorial density is sig-

nificantly less than the value of the restoring density there

due to equatorial upwelling).

In the version of the LOM used in this study, effects of

temperature and salinity on density are not separated. In

other words, the LOM prognostic tracer equation is for

density itself (see the appendix), which is a corollary of

temperature and salinity equations and a linear equation

of state for seawater. Consequently, we use the terms

pycnocline and thermocline and ocean density and ther-

mal structure interchangeably throughout the paper. Note

also that compressibility effects are not included in our

model, so it is appropriate to compare the density fields

that we plot to observations of potential or neutral density.

We include a small region with a higher restoring density

near the southern boundary (Fig. 2c) to generate a local-

ized region where deep convection occurs. This feature is

included because oceanic deep convection is localized in

nature; its impacts are discussed in section 3g. In pre-

liminary experiments we also included a similar pertur-

bation to the restoring density in the north, but found that

it had little effect on either stratification or overturning.

c. Spinup procedure and simulation characteristics

In addition to the featured simulation of an ocean with

zero diffusivity, we carry out several additional model

FIG. 1. Basin geometry: (a) north–south vertical cross section and

(b) horizontal (view from above). Note that the basin has sloping

boundaries on all sides as shown by the dotted and dashed lines in (b).


runs aimed at exploring the sensitivities of our solutions.

Most of these simulations start with the same initial spinup

in which the surface forcing, shown in Fig. 2, is applied to

an ocean that has a constant density of 1027 kg m23. This

is a somewhat arbitrary choice for an initial density, but

our preliminary experiments (not shown) suggest that the

quasi-steady circulations that ultimately develop are not

sensitive to the initial density, provided that it is less than

that of parcels sinking in deep convecting regions near the

poles. If the initial density is greater than the maximum

density of the restoring function, bottom water remains

unventilated in the nondiffusive case.

FIG. 2. Surface forcing: (a) zonal wind stress and (b),(c) restoring density. For restoring density we use a zonally

uniform profile shown as a solid line in (b) but modified near the southern boundary to include a localized source of

dense water at about 708S, 308W, evident in (c). Observed surface density along 308W is shown with a dashed line in (b).


Note that the initial stratification develops as a result

of nonentraining convective plumes parameterized via

the convective parameterization Lagrangian overturning

(LO), which is described in the appendix. That is, in lo-

cations where the surface restoring density generates

higher densities than the initial condition, vertical posi-

tion swaps of parcels associated with LO cause relatively

dense parcels to descend. This is a physical process; in

nature convective plumes transport relatively dense wa-

ter downward. Although there is mixing in these plumes

in nature, it is a matter of debate as to how much mixing.

Our representation of convection corresponds to the zero

entrainment end of the spectrum, which seems appro-

priate for a simulation of a nondiffusive ocean.

For the first 3000 years the model is run in its most

computationally efficient manner, which means parcels

divide (merge) when they enter a region of higher (lower)

vertical resolution. After 3000 years of spinup the circu-

lation and density fields are nearly in a statistically steady

state: the only nonsteady aspect of time-averaged fields is

a very weak increase in the density of the abyss with time

(,0.01 kg m23 per 1000 yr) that is decreasing in ampli-

tude with time. It is important to note that, whereas dif-

fusivity is set to zero in the main experiment, nonzero

vertical and horizontal viscosities are included (1023 and

7 3 104 m2 s21 respectively), which allows for energy

dissipation in the system and corresponds to an infinite

Prandtl number.

1) FEATURED SIMULATION: AN OCEAN WITH

ZERO DIFFUSIVITY

For the featured simulation, after 3000 years of spinup

we discontinue the dividing and merging of parcels and

run the simulation out 1000 years longer, without any

other modifications in model parameters. We make this

change because the occasional merging of parcels used in

the spinup generates a very small amount of tracer diffu-

sion, and we are interested in exploring the adiabatic limit

(i.e., in having a tracer diffusion of exactly zero) in this

study. When no merging of parcels is used, a water parcel’s

density is exactly conserved when it is not in contact with

the surface layer.

For example, Fig. 3 shows the latitude, depth, and

density of a water parcel during the last 70 years of the

nondiffusive simulation. Each curve is drawn as a solid

line for time segments in which the parcel has no contact

with the surface. During these segments there is abso-

lutely no change in density; that is, there are no com-

mands in the model code (other than surface forcing) that

alter the double-precision variable that represents the

parcel’s density, which is also confirmed by inspection of

model output files. Note that, although one of these pe-

riods is a deep and long trip through the deep western

boundary current (years 50–70), during other periods

without contact with the surface the parcel is never more

than a few hundred meters deep (e.g., years 20–35).

It turns out that switching to a uniformly high vertical

resolution does very little to change the stratification or

circulation characteristics. In fact, it appears that after this

entire 4000-yr period ocean adjustment toward a statisti-

cally steady state is nearly completed. This is confirmed

by integral measures of the adjustment, including effects

of the surface density forcing, the total kinetic energy, and

the average density below 2 km, along with their de-

rivatives (Fig. 4). Systematic variations in the integrated

effects of surface density forcing decay rapidly after the

simulation is started (Fig. 4a). Most of the kinetic energy

adjustment occurs in the first few hundred years of the

simulation, and then only relatively weak variations are

present, presumably associated with meanders or other

small variations in western boundary currents and the

circumpolar current (Fig. 4b). Similarly, most of the ad-

justment in deep stratification (Fig. 4c) occurs in roughly

the first 700 years of the nondiffusive simulation, with

only a very weak increase in the density of the abyss at

later times (less than 0.01 kg m23 per 1000 yr after ad-

justment to the new resolution has occurred).

Note that a spinup period of several thousand years

would be sufficient to expect a near steady state in a model

run with a typical vertical diffusivity of 1 cm2 s21, which is

FIG. 3. Evolution of (a) latitude, (b) depth, and (c) density for an

arbitrary water parcel during the last 70 yr of the nondiffusive

simulation. Time segments in which the parcel has no contact with

the surface are denoted with solid lines. Inspection of raw model

data reveals that there is absolutely no change in density during

each of these segments.


based on the appropriate diffusive time scale. However,

there is no comparable time scale for the nondiffusive run.

Rather, in this latter case the time scale of the adjustment

relates to how fast the abyssal ocean fills with dense water

that forms near the poles.

To illustrate the time scale of ocean ventilation we

present a cumulative distribution of transit times for water

parcels at the end of the nondiffusive simulation (Fig. 5a).

The transit time for each parcel is defined as the time from

the parcel’s last contact with the surface to its current

position, which is shown in Fig. 5a (solid line) for the end

the nondiffusive simulation (4000 yr). We see that roughly

70%–80% of the parcels have had contact with the surface

in the last 1000 yr (Fig. 5a). In contrast, much of the

bottom water (z . 3000 m) in the Northern Hemisphere

requires more than 3000 yr for ventilation (Fig. 5b). When

tracer diffusion is included, ventilation of this bottom

water occurs at a much more rapid pace (Fig. 5a, dotted

line; Fig. 5c).

Note that the transit time of a water parcel is a natural

and simply defined concept for a Lagrangian model: that

is, the time that has elapsed since the parcel was last at the

surface. For a nondiffusive ocean the transit time (as

defined above) and the water ventilation age are synon-

ymous. However, for a diffusive ocean the two will differ

because the ventilation age of a submerged parcel is an

integral quantity that depends on the parcel’s transit time

and the effects of mixing with surrounding parcels. Since

most of this study deals with a nondiffusive ocean, here

we do not distinguish the parcel’s transit time and venti-

lation age.

2) THE DIFFUSIVE OCEAN AND OTHER

SIMULATIONS

Besides the featured nondiffusive simulation, we carry

out an additional simulation to examine how ocean cir-

culation and stratification change when tracer diffusion

is included. The diffusive simulation uses the same reso-

lution as the spinup run but includes a tracer diffusiv-

ity value varying linearly from 1.2 cm2 s21 at a 4.9-km

depth to 0.3 cm2 s21 at the surface. Comparing density

and flow fields for this simulation to those for the non-

diffusive ocean illustrates the contributions of tracer dif-

fusion to stratification and meridional overturning and

how key water pathways are altered by the presence of

diffusion. In addition, we carry out several experiments to

test the sensitivity of overturning, stratification, and bolus

transports to changes in the horizontal and vertical res-

olution of the model (see the appendix and section 3e)

and to examine the impacts of the localized perturbation

to the restoring density in the southern high latitudes on

ocean stratification (section 3g).

3. Stratification and circulation in a nondiffusiveocean

In this section we examine circulation and density struc-

ture in an ocean with no tracer diffusion. Water parcels

exactly conserve their densities when they are not in con-

tact with the surface layer.

a. Surface features and horizontal circulations

The near-surface density and flow patterns for the run

with zero interior diffusion are shown in Fig. 6a. Prom-

inent features include western boundary currents, Ekman

divergence over the equator with a weak dense (cold)

tongue, a meandering circumpolar current, and an east-

ern boundary current in the northern subpolar region. As

expected, boundary currents deflect isopycnals in the di-

rection of the flow and, overall, density and flow patterns

are not that different from typical density-diffusing

simulations for idealized basins of this kind. Note that,

FIG. 4. Integral measures of adjustment toward a steady state: (a)

average effective volume flux (EVF) resulting from density re-

storing at the ocean surface, (b) ocean total kinetic energy, and (c)

average density below 2 km. In each panel, the average value of the

stated variable over 40-yr time segments is plotted with a solid line,

and its derivative with respect to time with a dashed line.


although the eastern boundary current in the northern

subpolar region causes a strong deflection in isopycnals,

this is partly attributable to the weaker gradient in the

restoring density north of 608N (Fig. 2b); this current is

actually much weaker than the western boundary cur-

rent in the northern subtropics (Fig. 6a).

The horizontal streamfunction is shown in Fig. 6b.

Circulations resembling classical Munk gyres appear

in the Northern Hemisphere with a much weaker sub-

tropical gyre and a strong circumpolar current in the

southern portion of the basin. The Northern Hemispheric

western boundary current transports roughly 35 Sv

(Sv [ 106 m3 s21), and the circumpolar current has an

amplitude of about 120 Sv, both of which are broadly in

line with their counterparts in the Atlantic Ocean (e.g.,

Cunningham et al. 2007; Nowlin and Klinck 1986). The

meanders of the circumpolar current suggest steering by

bottom topography (e.g., Killworth and Hughes 2002).

The Southern Hemispheric subtropical gyre appears

to be weaker than that in the north for two reasons:

1) the northward and southward flows in the deep over-

turning cell occur at different longitudes, and they en-

hance the western portion of the streamfunction for the

Northern Hemispheric subtropical gyre but weaken the

streamfunction for the western portion of the southern

hemispheric subtropical gyre (Figs. 6b–d) and 2) the

steady meander of the circumpolar current interferes

with the flow of the subtropical gyre in the Southern

FIG. 5. (a) Cumulative distribution of transit times for all water parcels in nondiffusive and

diffusive simulations (solid and dashed lines, respectively) at the end of each model run. The

transit time is defined as the time it takes for a parcel to travel from the surface to its current

position. Each parcel is marked with a shaded box to indicate its transit time for (b) non-

diffusive and (c) diffusive simulations. This figure demonstrates that in the diffusive run a vast

majority of the parcels have been in contact with the surface within the last 1000 yr. In contrast,

in the nondiffusive run bottom water north of the equator is only very weakly ventilated, if at

all, with many transit times exceeding 3000 yr.


Hemisphere. These points are illustrated by plots of

transport and thickness for the 1025–1026 kg m23 and

1027–1028 kg m23 density layers (Figs. 6c,d). The former

shows flow and thickness perturbations associated with

subtropical gyres (Fig. 6c), and the latter shows the po-

sitioning of the deep western boundary current (Fig. 6d),

which maintains a nearly a constant transport from 508N

to 508S.

FIG. 6. Surface fields and horizontal flow for the nondiffusive ocean averaged for the last 30 years of calculations:

(a) near-surface density (kg m23) and velocity, (b) barotropic streamfunction (contour interval 7 Sv), and layer

thickness and transport for (c) the 1025–1026 kg m23 density layer and (d) the 1027–1028 kg m23 density layer. Note

the equatorial cold tongue in (a) and the gyre systems and the ACC in (b). A branch of the ACC meanders northward

and then back at about 608W reminiscent of the Falkland (Malvinas) current.


b. Ocean density (thermal) structure

The subsurface ocean density structure in the non-

diffusive ocean is not unlike that observed in the Atlantic

Ocean (Fig. 7a), despite the idealized nature of the

model. Each of the major water masses in the Atlantic

is represented: warm surface tropical and subtropical

waters stratified according to the ventilated thermo-

cline theory (Luyten et al. 1983), Antarctic Intermediate

Water (AAIW) that subducts between 408 and 608S;

North Atlantic Deep Water (NADW; or ‘‘deep water’’)

that forms north of 408N; and Antarctic Bottom Water

(AABW; or ‘‘bottom water’’) that sinks south of 608S

(Fig. 7b). The main thermocline is about 800–900 m

deep in most of the basin, and its overall shape is similar

to that for the Atlantic, which is shown with the s 1 5 31.9

contour line in Fig. 7a. In much of the ventilated ther-

mocline, the main thermocline, and the deep ocean, sim-

ulated isopycnals run parallel to observed isopycnals. The

tropical thermocline, which exhibits a much greater den-

sity gradient, reaches depths of about 250 m (Fig. 7a).

Because each water parcel’s density is only altered by

surface forcing, it is essentially determined by the latitude

fs at which the parcel last had contact with the surface. In

Fig. 7b, we color code parcels according to their value of

fs, which is found by following reverse trajectories for the

last 700 years of the nondiffusive simulation. Simply by

comparing Figs. 7a,b, one can explain most of the density

structure in the idealized ocean. Water with a density of

less than 1026 kg m23 forms at latitudes between 408S and

408N and circulates within shallow tropical/subtropical

cells. Most of the water with densities between 1026 and

1027 kg m23 forms between 408 and 608S, and it seems

appropriate to refer to this water as Antarctic Intermediate

Water. North Atlantic Deep Water forms north of 408N

and spans depths of 1–3 km in most of the ocean. Finally,

Antarctic Bottom Water forms south of 608S and spreads

northward, slowly filling the ocean below about 3 km.

Note that Fig. 7b compares favorably with the observed

salinity structure in the Atlantic (Fig. 7c), which suggests

similar distributions and regions of formation for each of

the major water masses in the Atlantic Ocean. In par-

ticular, the model generates a northward intrusion of

AAIW that divides warm tropical and subtropical waters

from deep water (Fig. 7b), similar to that in the obser-

vations (Fig. 7c).

Also note that in Fig. 7b we deliberately truncate

trajectories at 700 years in order to illustrate the time

scale of ventilation for bottom water. Although most of

the bottom water in the Southern Hemisphere is ven-

tilated on a time scale of less than 700 yr, most of the

bottom water in the Northern Hemisphere is much

older (Fig. 5b).

To examine the ocean thermocline structure more

closely, we consider profiles of density for several loca-

tions and compare them to observations (Fig. 8). There

are several clearly defined thermoclines in the non-

diffusive ocean (e.g., Figs. 8a,b): the strong, upper-ocean

ventilated thermocline centered at about 200-m depth;

another (internal) thermocline located at the base of

mode water and centered around 800 m; and a weaker

internal thermocline separating NADW and AABW.

The depth of this thermocline is about 2900 m in the

middle of the basin and 2100 m near the basin southern

boundary, which matches the depth of Drake Passage.

The structure of the upper two thermoclines is some-

what similar to that discussed by Samelson and Vallis

(1997) and Vallis (2000). In their results the main ther-

mocline also effectively splits into two parts: the upper-

ocean ventilated thermocline that outcrops on isopycnals

in the subtropical gyre and a deeper internal thermocline

controlled by diffusion. In contrast, in our nondiffusive

ocean this internal thermocline is a nondiffusive bound-

ary layer between AAIW and NADW (Fig. 7b).

Overall, the comparison to observations (gray lines in

Figs. 8a,c,e and gray boxes in Figs. 8b,d,f) reveals the same

gross vertical structure of density for the nondiffusive

ocean, except for a light density bias of 0.3–0.4 kg m23.

The light bias results from a significant (negative) de-

parture of the idealized restoring density from the ob-

served surface density in low to midlatitudes (Fig. 2b)

combined with the overly idealized representation of deep

convecting regions.

c. Meridional overturning

A direct examination of the meridional overturning

circulation (Fig. 9) reveals that the deep cell includes

about 13 Sv of sinking water north of 308N, and stream-

lines suggest that almost all of this water upwells in the

Southern Ocean (Fig. 9a). There are also shallow, wind-

driven tropical/subtropical cells and a Deacon cell (Doos

and Webb 1994), a largely adiabatic overturning in the

Southern Ocean with a local maxima in circulation at

518S, 400 m.

Examining overturning with density as the vertical

coordinate (Fig. 9b) yields a higher-resolution view of

the shallow wind-driven cells, causes the Deacon cell to

disappear as expected, and illustrates how water par-

cels that traverse the subsurface portion of the deep

overturning cell (e.g., the deep western boundary cur-

rent) maintain a constant density, with flat streamlines

in the lower part of Fig. 9b. Overturning of bottom

water is very weak [O(1 Sv)] and it barely registers in

the lower-left region of Fig. 9b (note that, in our ide-

alized ocean, bottom water has densities around or

greater than 1027.5 kg m23, as shown in Figs. 7a,b).


d. Lagrangian analyses of water pathways

One advantage of our Lagrangian model is that it

provides precise trajectories for every water parcel in the

ocean with no added computations, allowing individual

mass elements to be tracked over long periods of time. In

this section, we use this information to track circulation

patterns for each of the major water masses identified in

Fig. 7b. In particular, we address the following questions:

where does sinking water ultimately upwell, and which

water pathways make the largest contributions to the

meridional overturning streamfunction?

FIG. 7. Ocean density and water masses. (a) Simulated density along 308W (colored con-

tours), along with observed isopycnals (dashed gray lines) for s0 5 26, 26.7; s1 5 31.9; and

s 3 5 41.45 (where sn is potential density referenced to n 3 1000 dbar). Note the upper-ocean

ventilated thermocline and an internal thermocline separating AAIW from NADW. (b) Water

parcels shown as small squares colored according to the latitude at which they last had contact

with the surface (showing different water masses). Black contours mark the 1026, 1027, and

1027.47 kg m23 isopycnals. Parcels shaded gray were not ventilated during the last 700 years of

the simulation; bottom water north of the equator primarily comprises such parcels. (c) Ob-

served (Levitus) salinity (psu) along 308 W. Note that the LOM produces a northward intrusion

of AAIW similar to that in nature.


1) SUBDUCTION AND UPWELLING OF NORTHERN

HEMISPHERIC WATER

In this section, we consider the fate of water parcels

that subduct in the northern portion of the ocean basin.

We examine trajectories during the last 70 years of the

model run and select paths for all parcels that sink from

the surface north of 308N and upwell south of 108N.

It turns out that there are essentially two key pathways

that satisfy these criteria, which we distinguish ac-

cording to the latitude where the water subducts. Be-

cause plotting many trajectories at one time yields a

‘‘spaghetti’’ diagram that is hard to interpret, we instead

objectively identify pathways most frequented by water

parcels that satisfy given upwelling and downwelling cri-

teria. One important question to which this information

provides a clear-cut answer is, where do parcels that sink in

the Northern Hemisphere upwell?

Figures 10a,c illustrate the preferred pathway to

upwelling for water that sinks north of 408N in two

coordinate systems. To construct a composite trajec-

tory, we divide longitude/latitude and latitude/depth do-

mains into 38 by 38 and 38 by 300 m bins, respectively,

and contours indicate the percentage of trajectories that

FIG. 8. Profiles of density and Brunt–Vaisala frequency as a function of depth for different locations for the

nondiffusive (black lines) and diffusive (dotted lines) simulations at (a),(b) 308N, 108W; (c),(d) 308N, 308W; and

(e),(f) 558S, 308W. Observed potential densities (gray lines) and Brunt–Vaisala frequencies (gray boxes) at the same

latitudes in the middle of the Atlantic are also shown. Note the ventilated thermocline near the surface, the first

internal thermocline centered at about 800-m depth [in (a)–(d)], and the second internal thermocline that separates

NADW and AABW in the nondiffusive experiment. The depth of the second internal thermocline is about 2900 m in

low latitudes and 2100 m in the southern high latitudes. The 0.3–0.4 kg m23 offset between modeled and observed

density profiles is a consequence of the idealized form of the restoring density (see text).


satisfy the given upwelling and downwelling criteria that

pass through a given bin. For example, a contour value of

50% in Fig. 10a indicates that half of the parcels that sink

north of 408N and upwell south of 108N during the last

70 yr of the simulation pass through a particular 38 by 38

region. This analysis reveals that parcels travel southward

in the deep western boundary current (Fig. 10a), reaching

depths from about 1 to 3 km between 408S and 408N

(Fig. 10c) before upwelling in the Southern Ocean.

The preferred location of subduction is where the deep

western boundary current starts in our model, that is,

where dense water in the subpolar gyre plunges beneath

relatively light water traveling northward in the near-

surface western boundary current associated with the

subtropical gyre (Fig. 6). Only a handful of parcels upwell

at the equator, and all of these sink at latitudes between

408 and 458N and remain less than 1 km deep. This result

illustrates that without tracer diffusion deep water cannot

contribute to the equatorial upwelling.

In contrast, all of the water parcels that sink from the

surface between 308 and 408N upwell near the equator

(Figs. 10b,d), with a much shallower subsurface pathway

and downwelling locations concentrated in the eastern

portion of the basin. The existence of distinct, well-

defined subsurface pathways, like those illustrated in

Figs. 7b and 10, is what leads to distinct regions of strat-

ification in Fig. 7a, and examining such pathways helps

one to understand how stratification develops in the

nondiffusive ocean. Note that the pathway illustrated in

Figs. 10b,d is generally consistent with ventilated ther-

mocline theory (Luyten et al. 1983; Pedlosky 1996), as is

the stratification shown in Fig. 7a.

Of course, the pathways in this model are much simpler

than those of water parcels in nature owing to the ideal-

ized basin and low resolution (e.g., the lack of mesoscale

eddies); yet, the model reproduces much of the density

structure and the basic water mass distribution observed

in the Atlantic Ocean, which validates our modeling ap-

proach (for a further discussion of this point, see section

4). Also note that our method of compositing illustrates

the most rapid and direct paths to upwelling; slow and/or

meandering paths to upwelling that take more than 70 yr

to complete are not included in the composite trajectories

shown in Fig. 10.

2) WATER THAT SUBDUCTS IN THE SOUTHERN

HEMISPHERE

In the Southern Hemisphere, the pathway to upwelling

for water that subducts in the shallow wind-driven cell of

the ventilated thermocline is similar to its counterpart in

the Northern Hemisphere, although in the southern cell

some of the water also upwells near the western boundary

(not shown). However, the fate of sinking water poleward

of the tropical/subtropical cell is quite different (Fig. 11)

and warrants a discussion.

Water that subducts between 408 and 608S (i.e., Antarctic

Intermediate Water) travels northward along the western

boundary at depths around or less than 1 km (Figs. 11a,c).

Some of it enters the equatorial undercurrent and upwells

in the eastern portion of the basin, but most continues

northward along the western boundary and upwells in

a zonally broad region near 458N.

Water that sinks poleward of 608S circulates in the

lower portion of the Deacon cell (Figs. 11b,d), with very

little penetration north of the equator, with upwelling

occurring in the Southern Ocean. Accordingly, the path-

way depicted in Figs. 11b,d describes parcel trajectories

within the adiabatic Deacon cell. Thus, this pathway is

clearly distinct from the classical bottom cell, which has

a much longer time scale and is very weak in the non-

diffusive ocean.

3) PARTITIONING THE MERIDIONAL

OVERTURNING CIRCULATION

Comparing the subsurface pathway to upwelling for

North Atlantic Deep Water (Fig. 10c) with the meridional

FIG. 9. Meridional overturning (contour interval 3 Sv) in (a)

height and (b) density coordinates for the nondiffusive simulation.

In (b) contours of average depth are for 300 (solid gray) and 1200 m

(dotted gray). Note that streamlines are flat in the lower part of (b),

indicating no density change in the lower branch of the deep over-

turning cell. The shallow subtropical cells are especially prominent

in (b). The Deacon cell is prominent in (a) but absent from (b). The

weak cell in the bottom-right corner of (b) indicates recirculation in

the subpolar gyre. The cell in the bottom-left corner of (b) is ap-

parently mostly a consequence surface density restoring along the

boundary of the channel. Less than 1 Sv of the volume transport of

this cell participates in ventilating the bottom ocean.


overturning streamfunction (Fig. 9a) brings up the fol-

lowing question: Why is the composite pathway to up-

welling (Fig. 10c) much more shallow than the streamlines

for the deep cell (Fig. 9a) in the southern portion of the

basin? To address this question, and others like it, we

seek a means of partitioning the meridional overturning

circulation into components associated with particular

water pathways. However, in general, arbitrary water

pathways are not mass balanced and they do not have

closed streamfunctions associated with them. To address

this challenge we define a ‘‘complete path’’ as a parcel

trajectory for which the starting and ending points have

the same latitude. Note that, when a parcel traverses

a complete path, there is zero vertically and temporally

integrated meridional transport at every latitude. Two

examples of complete paths are shown in Figs. 12a,c, one in

which the parcel traverses much of the deep overturning

cell (gray lines) and another in which a parcel remains

confined to the Southern Hemisphere (solid lines).

Next, we consider all the parcel trajectories for the last

700 years of the nondiffusive simulation and break each

into a series of complete paths and leftover incomplete

paths. For a given parcel, each complete path starts

and ends at the parcel’s initial latitude at the beginning

of the 700-yr period. In general, most parcels return to

their starting latitude multiple times, yielding multiple

complete path segments, but they typically do not end

up at the same latitude at which they started. Therefore,

the last leg of their journey is usually an incomplete path.

We hypothesize that, to the extent that the incomplete

paths represent a random sample of ocean circulations,

excluding them will not change the structure of the over-

turning but reduce its amplitude. Indeed, we find that, by

multiplying the streamfunction for all complete paths by

1.55 (not shown), we are able to obtain the gross struc-

ture of the streamfunction for all paths (Fig. 9a). More-

over, the complete path streamfunction can be partitioned

in an arbitrary manner, yielding closed streamfunctions

for each component of the partition. For example, Fig. 12b

shows the streamfunction associated with all complete

paths that remain south of 208N, and Fig. 12d shows the

streamfunction for all complete paths that pass north of

208N at some point.

It is clear from these figures that the plunging of the

streamlines toward the bottom in Fig. 9a is not associated

with the classic, bottom overturning cell. Rather, it is par-

cels that meander northward and southward at different

depths as they traverse the circumpolar current (e.g.,

FIG. 10. Pathways to upwelling for water that subducts in the Northern Hemisphere in the nondiffusive ocean. (a),(c) Composite

trajectory for all parcels that sink north of 408N and upwell south of 108N during the last 70 yr. (b),(d) Composite trajectory for all parcels

that sink between 308 and 408N and upwell south of 108N during the last 70 yr. In each panel, black and white contours indicate the

percentage of trajectories that pass within 38 and/or 300 m of a given location, and shading indicates the percentage of parcels that

downwell (blue) or upwell (red) within 38 and/or 300 m of a given location. The duration of different trajectories ranges from 12 to 69 yr

for (a),(c) and from 2 to 55 yr for (b),(d).


Figs. 12a,c, solid lines) that cause the very deep over-

turning in the Southern Hemisphere shown in Fig. 9a.

Thus, the circulation depicted in Fig. 12b is essentially

the Deacon cell for our idealized basin (Doos and Webb

1994). It results from the corkscrewing motion of a por-

tion of the circumpolar current that occurs in response

to variable bottom topography. We have found that this

cell becomes shallower (i.e., more like the Deacon cell

in nature) in preliminary experiments that use realistic

topography instead of the idealized basin.

Our Lagrangian analysis and conclusions are largely

consistent with those of Doos et al. (2008). However, in

that study there were more water pathways contributing

to the Deacon cell owing to the use of realistic topogra-

phy and a global domain, and trajectories were truncated

at 308S, which may have reduced the contributions at-

tributed to the circumpolar current.

When pathways that are confined to the south are

removed from the total streamfunction, the residual

streamfunction (Fig. 12d) becomes consistent with the

pathway to upwelling for the deep cell (Fig. 10b). This

residual streamfunction is analogous to the residual-mean

circulation obtained in a zonally averaged model of the

Atlantic meridional overturning circulation (AMOC) by

Sevellec and Fedorov (2011).

e. Bolus transports

One reason that our idealized ocean contains realistic

density and overturning structures is that, even when run

at a low resolution, the LOM spontaneously generates

eddylike bolus transports. When a conventional ocean

model is run at a coarse resolution, the GM parameteri-

zation is typically used to represent the transport of iso-

pycnal layer thicknesses by unresolved eddies (e.g., Bryan

et al. 1999). However, we have found no need to include

such a parameterization in the LOM as the model spon-

taneously generates such a transport with approximately

the right magnitude.

For example, Fig. 13 shows the meridional transport of

the 1026–1027 kg m23 layer thickness by transient eddies

(TE, solid lines) compared with what a GM parame-

terization would generate (GM, dashed lines) for a dif-

fusion constant of 500 m2 s21 for 38 and 1.58 resolutions

(Figs. 13a,b). The TE and GM transports are highly cor-

related, and the TE transport is similar at the two different

resolutions. For the higher-resolution run, the TE transport

FIG. 11. Pathways for water that subducts in the Southern Hemisphere in the nondiffusive ocean. (a),(c) Composite trajectory for all

parcels sinking between 408 and 608S and upwelling north of 108S in the last 70 yr of the nondiffusive simulation. (b),(d) Composite

trajectory for all parcels sinking south of 608S and advancing north of 308S and to depths greater than 3 km prior to upwelling in the last

700 yr of the nondiffusive simulation. Contouring and shading as in Fig. 9. The duration of different trajectories ranges from 6 to 66 yr for

(a),(c) and from 6 to 166 yr for (b),(d).


is slightly weaker in the vicinity of the Antarctic Circum-

polar Current (ACC); however, in the higher-resolution

run, the ACC exhibits stronger meanders (cf. Figs. 13c,d),

suggesting a greater transport by standing eddies and

opening up the possibility of eddy transports across the

polar front in the zonal direction (evident at even higher-

resolution runs but not shown here).

Rigorously explaining how the LOM generates these

transports goes beyond the scope of this paper, but we do

mention two key factors: 1) unlike in conventional models

in which grid points are aligned in columns, in the LOM

parcel centers are randomly distributed, leading to a bet-

ter coverage of the horizontal domain, and 2) individual

parcels can take paths that depart from the mean flow and

are more like water pathways in nature and/or higher

resolution runs. For an example of the latter, compare the

parcel trajectory shown by the solid line in Fig. 12a with

the mean flow of the ACC as indicated by Figs. 6b, 13c.

The parcel makes a sharp and extensive northward loop

(Fig. 12a), much more pronounced than that in the mean

flow of the ACC (Figs. 6b, 13b) and more like the mean

flow of the higher resolution run (Fig. 13d). Finally, there

are hints that the generation of bolus transports in the

LOM may have similarities to the horizontal mixing by

chaotic Lagrangian transport described by Rogerson et al.

(1999) and Yuan et al. (2004).

f. Effects of including tracer diffusion

Although the focus of this paper is on the circulation and

density structure that develop in an ocean with zero dif-

fusivity, it is also instructive to include a realistic value for

diffusivity to see how circulations and stratification change:

that is, to pin down exactly what tracer diffusion contrib-

utes to the problem. In this section, we present a simula-

tion that includes a vertical tracer diffusivity that varies

linearly from 0.3 cm2 s21 at the surface to 1.2 cm2 s21 at

4.9-km depth (similar to the values used by Toggweiler and

Samuels 1998). The model is run for 3000 years with this

diffusion, which is more than enough time to for the ocean

to reach an equilibrium.

Figure 14 shows the density structure and the latitude

at which parcels last had contact with the surface for the

idealized ocean with tracer diffusion. Comparing this figure

to Fig. 7 reveals several important changes caused by the

presence of tracer diffusion. First, while the density of

the upper ocean does not change much (Figs. 14a, 7a, 8),

the main thermocline becomes less sharp. Second, diffu-

sion decreases both the density and stratification of the

FIG. 12. Partitioning of the meridional overturning circulation into cross-hemispheric and Southern Hemispheric

components: (a),(c) two examples of complete paths. The starting point of each path is marked with a shaded square.

Note that the black path remains south of 208N, the gray path passes north of 208N, and each path finishes at the same

latitude at which it starts. (b) Overturning streamfunction associated with all complete paths that remain south of

208N during the last 700 yr of the nondiffusive simulation, which shows the model’s equivalent of the Deacon cell

(3-Sv contour interval). (d) Overturning streamfunction associated with all complete paths that pass north of 208N at

some point during the last 700 years of the nondiffusive simulation (contour interval 3Sv), which shows the deep cell

isolated from the Deacon cell. In effect, this shows the model residual-mean circulation. The streamfunctions in

(b),(d) are scaled to make up for the missing contribution of incomplete paths by multiplying by 1.55.


abyssal ocean, which allows North Atlantic Deep Water

to reach greater depths (as it does in nature). Third, al-

though Fig. 14b suggests that each of the major water

masses are transported downward and horizontally in

a similar manner to that indicated by Fig. 7b, when tracer

diffusion is present the boundaries between water masses

become less distinct. For example, in Fig. 7a, AAIW re-

mains largely isolated from water masses above and be-

low it well to the north of the equator, but in Fig. 14b

there is a substantial intrusion of North Atlantic Deep

Water into the AAIW in the tropics.

We more precisely quantify density and stratification

changes caused by vertical tracer diffusion by examining

vertical profiles of density and Brunt–Vaisala frequency

for several locations for both diffusive and nondiffusive

runs (Fig. 8). We see that including diffusion does not alter

the density of the upper ocean much (Figs. 8a,c,e); rather,

it simply ‘‘rounds off’’ the inflection point in density near

the bottom of the main thermocline. In other words, dif-

fusion makes the internal thermocline at the base of the

mode water less sharp. However, this change significantly

affects the vertical gradient in density and the Brunt–

Vaisala frequency, making it more realistic (Figs. 8b,d,f).

Moreover, the second internal thermocline formed at the

boundary between North Atlantic Deep Water and Ant-

arctic Bottom Water (Figs. 8b,d) in the nondiffusive ocean

disappears when diffusion is included.

Although including diffusion increases the amplitude

of the deep ocean overturning cell by several sverdrups

(cf. Figs. 15a, 9a), the bulk of the transport and the overall

structure of the cell is reproduced in the adiabatic limit.

Diffusion also generates a realistic bottom overturning

cell (Fig. 15a), leading to much more rapid ventilation of

bottom water in the Northern Hemisphere (Figs. 14b, 5),

which is more consistent with observations (e.g., Orsi

et al. 2001).

FIG. 13. Northward bolus transport of 1026–1027 kg m23 layer thickness for (a) 38 and (b) 1.58 model resolutions in

nondiffusive runs (solid lines). The bolus transport is calculated by first calculating a time-average layer thickness H

and meridional velocity y for each horizontal location and then computingÐ

y9H9 dx, where the prime notation

denotes perturbations, the overbar a temporal average, x is the zonal distance, and the limits of integration are 628W

and 28E. In each panel, the transport that a conventional GM parameterization would generate in this experiment is

shown with a dotted line. A GM coefficient of 500 m2 s21 was used. Time-averaged 1026–1027 kg m23 layer

thickness (100-m contour interval) and transport (vectors) for (c) 38 and (d) 1.58 resolutions. At the higher resolution,

the transient eddy transport is reduced slightly in the Southern Ocean; however, steady meanders in the circumpolar

current (i.e., standing eddies) become more intense. This figure demonstrates that, even with a relatively coarse

resolution, the LOM can reproduce a realistic bolus transport without explicit eddy parameterizations such as GM.


Finally, by converting density into temperature we es-

timate the oceanic northward heat transport for both the

nondiffusive (solid) and diffusive (dashed) simulations

(Fig. 16). Here, we have assumed that density variations

are proportional to temperature variations times a ther-

mal expansion coefficient of 0.2 kg m23 K21. This figure

suggests that the bulk of the heat transport is carried out

by adiabatic, wind-driven ocean circulation. In most of

the domain, the simulated heat transport in our non-

diffusive ocean is in the range of estimates for the At-

lantic Ocean based on observations (Fig. 16b; adapted

from Ganachaud and Wunsch 1993).

The estimated northward heat transport in the run with

diffusion is generally 10%–20% higher than that for the

nondiffusive ocean. This difference would have been even

smaller if the Pacific basin were included in consideration.

g. Effects of localized deep convection

As we note in section 2, we include a localized pertur-

bation to the restoring density near the southern bound-

ary to cause deep convection to be localized, as it is in

nature. In this section, we present the results of a 3000-yr

simulation identical to the spinup run except that it em-

ploys a zonally symmetric restoring density (Fig. 2b)

without the perturbation near the southern boundary.

The primary change associated with using this alternative

restoring density is a decrease in the density and stratifi-

cation of bottom water (Fig. 17a), which leads to a deeper

and stronger deep cell (Fig. 17b) and NADW penetrating

closer to the bottom (cf. Figs. 17c, 7b).

This run illustrates several key points. In general, it is

not necessary to include localized perturbations to the

restoring density profile to reproduce the gross stratifi-

cation, water mass distribution, and overturning structure

observed in nature. On the other hand, small perturba-

tions to the restoring density in the Antarctic can have

large impacts on abyssal stratification, so including tracer

diffusion is not the only way to obtain more realistic deep/

abyssal stratification.

4. Summary and discussion

In this study, we examine the circulation and density

structure in an idealized ocean with a diffusivity of zero.

FIG. 14. Ocean density structure and different water masses for the simulation with tracer dif-

fusivity ranging from 0.3 cm2 s21 at the surface to 1.2 cm2 s21 at 4.9 km: (a) density along 308W

(colored contours) along with the 1024, 1025, and 1026 kg m23 isopycnals for the nondiffusive

simulation (gray dashed lines). (b) Water parcels shown as small squares colored according to the

latitude at which they last had contact with the surface during the last 700 yr of the diffusive

simulation, which illustrates different water masses. Parcels shaded gray were not ventilated during

this time frame. Black contours mark the 1026, 1027, and 1027.47 kg m23 isopycnals.


The ocean basin is roughly the size of the Atlantic and it

includes an extended periodic circumpolar channel. We

apply a zonal wind stress and a surface density restoring

function that are analytic approximations of those ob-

served over the Atlantic. We also conduct a simulation

with identical surface forcing that includes a moderate

amount of interior diapycnal diffusion, as well as several

other simulations that test sensitivities to model reso-

lution and the structure of the restoring density.

With zero diffusivity, the idealized ocean develops

a stratification not unlike that seen in nature, which

amounts to a layering of water masses formed at different

latitudes. Each of the major water masses in the Atlantic

is represented: light subtropical and tropical waters in

a ventilated thermocline, Antarctic Intermediate Water

that forms between 408 and 608S and spreads northward

under subtropical waters, North Atlantic Deep Water

that forms north of 408N and occupies most of the ocean

at depths between 1 and 3 km, and Antarctic Bottom

Water that sinks south of 608S and fills depths greater

than 3 km.

The idealized ocean also develops a strong deep me-

ridional overturning that comprises two components: a

Deacon cell or adiabatic overturning associated with me-

andering and/or corkscrewing of a portion of the cir-

cumpolar current (Doos and Webb 1994; Doos et al. 2008)

and the classic deep cell in which water sinks in the North

Atlantic, travels southward in a deep western boundary

current, upwells in the Southern Ocean, and is converted

to AAIW before returning northward beneath the tropi-

cal/subtropical cells.

The deep overturning cell associated with the NADW

in our nondiffusive ocean (13 Sv) is somewhat weaker

than recent observations indicate (e.g., 18.7 65.6 Sv in

Cunningham et al. 2007). However, there are a number

of factors unrelated to diffusion that can account for the

difference. These include the relatively low vertical res-

olution of the model (a higher resolution increases the

overturning as shown in the appendix) and the specified

depth of the circumpolar channel (a deeper channel re-

sults in stronger overturning, as will be discussed else-

where). Thus, our simulations suggest that it is quite

possible that a nondiffusive ocean could support a deep

overturning within the error bars of that observed in the

Atlantic Ocean.

One intriguing aspect of the nondiffusive simulation

is that, despite the fact that major ocean currents are

underresolved and that water parcels are much larger than

mesoscale eddies, the gross density structure and water

mass distribution of the Atlantic Ocean is reproduced.

One reason for this result is that, even at very low reso-

lution, the Lagrangian model spontaneously generates

bolus transports of isopycnal layer thickness having an

amplitude and structure similar to that which a GM eddy

parameterization would produce. Moreover, due to its

Lagrangian nature, the model perfectly conserves every

FIG. 15. Meridional overturning (contour interval 3 Sv) for the

diffusive ocean in (a) height and (b) density coordinates. In (b)

contours of average depth are for 300 (solid gray) and 1200 m

(dotted gray). Note the well-developed bottom overturning cell as

well as upwelling of deep water induced by vertical mixing.

FIG. 16. A comparison of simulated and observed heat trans-

port: (a) oceanic northward heat transport for the nondiffusive

and diffusive simulations (solid and dashed lines, respectively)

and (b) a variety of estimates of northward heat transport for the

Atlantic Ocean adapted from Ganachaud and Wunsch (1993)

(refer see Fig. 3 of that study for details on these estimates).


moment of tracer distributions, so its low resolution

does equate with excessive spurious diffusion, as can be

the case for Eulerian models (Griffies et al. 2000).

We also note that major large-scale components of

ocean circulation appear to be adequately resolved, in-

cluding Ekman transport in the surface layer, equator-

ward Sverdrup flow in the interior of the ventilated

thermocline, and regions of light and dense water for-

mation at the surface. Is it possible, to the extent that

boundary currents compensate for imbalances gener-

ated by these large-scale features and that the gross

effects of mesoscale eddies are accounted for by bolus

transports, that the fine structure of currents and mesoscale

circulations is not of zero-order importance for the density

structure and water mass distribution in the oceans? In-

creasing the resolution of our nondiffusive ocean to the

mesoscale, which we hope to do in the future, might help to

address this important question.

A fundamental result of this study is that the nondiffu-

sive model generates a realistic northward heat transport in

the Atlantic (;1.2 PW). This result is consistent with the

conclusion of Boccaletti et al. (2005), who introduced ‘‘heat

FIG. 17. Results for the nondiffusive simulation with a zonally symmetric restoring density:

that is, lacking the perturbation near the southern boundary. (a) Density profile at 308N, 308W

(solid line). (b) Meridional overturning (contour interval 3 Sv). (c) Different water masses.

Colors indicate the location of each parcel’s last contact with the surface during the last 700 yr

of the simulation. The density profile for the original experiment (with a localized perturbation

to the restoring density) is shown with a dashed line in (c).


function’’ to argue that most of the ocean heat transport

is done by the shallow surface-intensified circulation.

The modeling results and Lagrangian analysis pre-

sented in this study provide a complimentary perspective

on the overturning circulation in the Southern Ocean to

that of the residual-mean theory of Marshall and Radko

(2003) and its extension in Sevellec and Fedorov (2011). A

Lagrangian transport pathway like that shown in Fig. 12d

develops because eddy transports cancel out much of

the Ekman transport in the Southern Ocean. Whereas

the studies of Marshall and Radko (2003) and Sevellec

and Fedorov (2011) invoked the effect of transient meso-

scale eddies, our Lagrangian analysis emphasizes the three-

dimensional nature of the flow and, in particular, the effect

of stationary eddies. Although there are bolus transports in

the LOM (Fig. 13) analogous to those typically attributed

to mesoscale eddies, their contribution is smaller than that

of the meander in the ACC, which is essentially a large

standing eddy (Figs. 6, 13). A recent high-resolution mod-

eling study by Ito et al. (2010) also suggests that transports

by standing eddies may do more to cancel Ekman transport

over the Southern Ocean than transient eddies.

The nondiffusive ocean develops several clearly defined

thermoclines. We note the strong ventilated thermocline

centered at about 200-m depth and an internal thermocline

at 800 m (Figs. 7, 8). This structure is somewhat similar to

that in the double-thermocline model of Samelson and

Vallis (1997) and Vallis (2000) for low-diffusion regimes.

Their internal thermocline, however, is controlled by dif-

fusion, whereas in our case it represents a nondiffusive

boundary layer between Antarctic Intermediate Water

and North Atlantic Deep Water. Therefore, our internal

thermocline is effectively ventilated as well, with a broad

region of source regions for the water contained in it. It is

noteworthy that at depths in the nondiffusive ocean a

weaker, second, internal thermocline is formed at the

boundary between NADW and Antarctic Bottom Water.

Vertical stratification between the two internal thermo-

clines remains very weak.

There are several notable features of the nondiffusive

ocean that are inconsistent with observations. For ex-

ample, the deep ocean below the main thermocline and

above roughly 3 km develops almost no stratification and

becomes separated from bottom waters by a weak but well-

developed pycnocline. Available observations do not sup-

port the existence of this second permanent pycnocline even

though it may still be possible in some regions of the ocean.

The bottom overturning cell in our nondiffusive sim-

ulation is also very weak, on the order of 1 Sv, and it

appears to be weakening with time. Whether the bottom

overturning cell will eventually disappear in the non-

diffusive experiment (on time scales longer than used in

this study) is unclear at this point. It is feasible that there

will remain a very weak bottom overturning cell related

to inherent model turbulence allowing individual parcels

from the ocean abyss to reach the surface on rare occa-

sions. This question, however, is not critical since with

higher diapycnal diffusion the strength of the bottom cell

increases to almost 7 Sv, which better agrees with ob-

servations for the Atlantic (e.g., Talley et al. 2003).

The weakness of the bottom cell in the adiabatic

simulation explains another property of the nondiffusive

ocean—by changing the restoring density near the basin’s

southern boundary (Fig. 2c) we can produce an arbitrary

vertical stratification in the abyss. In fact, using a weaker

localized density perturbation near the southern boundary

would reduce the simulated stratification in the abyssal

ocean to more realistic values (Fig. 17a) and allow a deeper

penetration of NADW into the ocean.

Despite the idealized bathymetry, the relatively coarse

model resolution, and the lack of atmospheric coupling,

the nondiffusive ocean reproduces the gross stratification,

water mass distribution, and meridional overturning ob-

served in the Atlantic Ocean. Therefore, we conclude that

surface forcing and adiabatic interior circulations funda-

mentally determine the leading-order density and circu-

lation structure for the ocean, whereas tracer diffusion

causes some important first-order perturbations including

a higher, more realistic, stratification just below the main

thermocline, an increase in the amplitude of deep over-

turning of several sverdrups, an increase in northward

heat transport in the Atlantic of 10%–20%, and a realistic

bottom overturning cell.

Acknowledgments. This research was supported in

part by grants from NSF (OCE-0901921), Department

of Energy Office of Science (DE-FG02-08ER64590),

and the David and Lucile Packard Foundation. We

thank George Philander, David Marshall, Bill Young,

Rui Xin Huang, Geoff Vallis, Joe Pedlosky and Taka Ito

for several discussions of this topic.

APPENDIX

The Lagrangian Ocean Model

a. Parcel shapes

The Lagrangian ocean model (LOM) represents a

body of water as a collection of conforming water par-

cels. An understanding of how the LOM works starts

with knowing the details of how water parcels are con-

structed. The vertical thickness H of each parcel satisfies

the following equation:

H(x9, y9) 5 Hmaxpjx9jrx

� �p

�jy9jry

�, (A1)


where Hmax is the parcel maximum vertical thickness, r x

and r y are the parcel radii in the x and y directions, the

prime notation denotes a coordinate system centered on

the parcel, and p is a third-order polynomial: p(x) 5 1 1

(2x 2 3)x2 for x , 1 and p(x) 5 0 for x $ 1. Figure A1a

illustrates the bell-shaped vertical thickness distribution,

and Fig. A1b shows how a collection of parcels that satisfy

(A1) can have a variety of shapes. The vertical positions

of parcels are determined by their ‘‘stacking order,’’

which is best visualized by thinking of placing the parcels

in the basin one at a time with a parcel’s rank in the

stacking order corresponding to its rank in the order of

placement.

Despite the complexity of the shapes shown in Fig. A1b,

it is relatively easy to define the parcel surfaces mathe-

matically. The top of the nth parcel is given by the fol-

lowing formula:

b 1 �n

i51Hi(x 2 xi, y 2 yi), (A2)

where b(x, y) is the height of the bottom topography and

(xi, yi) denotes the horizontal position of the ith parcel. In

other words, (A1) only constrains a parcel’s vertical

thickness; its shape is also determined by the height of its

lower surface, which depends on the height of the bottom

topography and the thicknesses of the parcels below.

We use parcel radii rx and ry that are fixed in time, which

means that a parcel’s vertical thickness function is not

distorted by the flow. While this aspect of the LOM’s

parcels is different from that of water parcels in the ocean,

it means that there is no need to remap water mass to

parcels after the initialization of the model, and individual

parcels can be tracked for thousands of years. Moreover,

because fluids are continuous (as we think of them) and

numerical representations of them are discrete, every

ocean model must have some sort of discretization along

these lines that has no counterpart in a real ocean.

b. Parcel density and maximum vertical thickness

The version of the LOM used in this study predicts

changes in a parcel’s density r using the following equation:

dr

dt5 Drsf 1 Drdiff , (A3)

where the subscripts ‘‘sf’’ and ‘‘diff’’ denote changes due to

surface forcing and tracer diffusion, respectively. The

surface forcing is a restoring to a prescribed surface density

with a piston velocity (Griffies et al. 2009) of about 1 m

day21. The implementation of tracer diffusion is discussed

briefly below and in more detail by Haertel et al. (2004,

2009). Note that for the featured simulation presented

here the rightmost term in (A3) is set to zero so that

a parcel’s density is exactly conserved when it is not in

contact with the surface. The amount of mass associated

with each parcel is fixed in time. Therefore, unlike models

using the Boussinesq approximation, the LOM conserves

the total mass, not the ocean volume, A parcel’s maxi-

mum vertical thickness is determined as follows:

Hmax 5M

rrxry

, (A4)

where M is the parcel mass. In other words, the model

predicts changes in density using (A3) and then diagnoses

Hmax using (A4). The LOM can also be configured to

predict changes in temperature and salinity independently

and to calculate density using an equation of state, but we

do not make use of these capabilities in this study.

c. Equations of motion

Horizontal motions of parcels are predicted using

Newtonian mechanics:

dx

dt5 v (A5)

and

dv

dt1 f k 3 v 5 A p 1 Am, (A6)

where x denotes horizontal position, v is horizontal ve-

locity, t is time, f is the Coriolis parameter, k is the unit

vector in the vertical, Ap is the acceleration resulting

from pressure, and Am is the acceleration resulting from

FIG. A1. Schema of how the LOM represents a body of water as

a collection of conforming parcels: (a) the shape of a single water

parcel when placed on a level surface and (b) a collection of water

parcels in an idealized basin, each with the same thickness dis-

tribution from its center as that shown in (a). The ocean is in a

hydrostatic equilibrium and no surface forcing is applied. The

thickness of each parcel as a function of distance from its center is

given by Eq. (A1). Adapted from Haertel and Straub (2010).


momentum exchange with nearby parcels (i.e., horizontal

and vertical viscosity). To simplify the calculation of Ap,

which is an integral of the inward normal vector times

pressure over the entire surface of a parcel, it is assumed

that pressure is hydrostatic and that density is spatially

uniform within a parcel, which yields the following formula:

Api5

1

Mi

ðg$Hi

"�

k

j5i11(rj 2 ri)Hj

1 ri b 1 �k

j5iHj

0@

1A#dm, (A7)

where the integral is evaluated over the horizontal pro-

jection of parcel i, g is gravity, k is the total number of

parcels, and dm is the horizontal area measure. We ap-

proximate (A7) with a Riemann sum, which leads to the

conservation of energy in the limit as the time step ap-

proaches zero and requires O(k) operations to evaluate

for k parcels (Haertel and Randall 2002).

In this study we use a form of gravity wave retardation

(Jensen 1996) that slows external gravity waves by a factor

of 10. Physically speaking, this is done by assuming that,

rather than displacing air, water parcels displace a fluid

with a density of 1017 kg m23, which is roughly 99% of the

average density of the ocean. This fluid is represented in

(A7) as the kth parcel, and it is assumed to have a perfectly

level free surface.

Equations (A5) and (A6) are solved at each time step

to find new locations for the centers of the parcels. Note

that in theory it would be possible for the parcels to

disperse, creating a void. However, in practice this does

not happen, because divergent flow creates a depression

in the free surface, which causes a pressure gradient that

accelerates parcels toward the center of the depression.

In other words, the pressure force acting alone tends to

produce circulations that level out the free surface, as is

the case for bodies of water in nature.

d. Convective parameterization

The LOM employs a representation of convection that

is unique to its Lagrangian framework. After each time

step parcel stacking orders are sorted by density so that

dense parcels lie beneath not so dense parcels. Figure A2

illustrates the effects of this convective adjustment scheme.

In the event that surface forcing or diffusion causes a local

unstable stratification (i.e., a dense parcel to lie above a

not so dense parcel), the vertical positions of the of-

fending parcels are swapped so that a stable stratification

is restored. We have explored the potential ramifica-

tions of this unique convective parameterization in a

Lagrangian model of the atmosphere and obtained very

encouraging results, with atmospheric convective systems

appearing to be more realistic than those generated by

conventional atmospheric climate models (Haertel and

Straub 2010).

e. Viscosity and diffusion

Viscosity and tracer diffusion are implemented in the

LOM in the following manner. First, parcel centers are

partitioned into rows and columns that run parallel to

each coordinate axis, where density is used for a vertical

coordinate. Then each row or column of parcel centers is

treated like a row or column of points in an Eulerian finite

difference model. In other words, a flux of momentum or

density is calculated between each parcel and its nearest

neighbors in each mixing row or column in which it is

contained. For more details on the implementation of

vertical and horizontal diffusion, the reader is referred

to Haertel et al. (2004, 2009), respectively.

f. Dividing and merging parcels

For computational efficiency, during the spinup simu-

lation parcels are moved in groups of two and four in the

middle (700–2100 m) and deep (.2100 m) ocean, respec-

tively. This is essentially the same as using a lower vertical

resolution in these regions. Because a single density is used

for each group, the joining of parcels into a group creates

FIG. A2. The convective adjustment scheme: locations of a collec-

tion of parcels within a model column are marked with boxes. Parcel

A is less dense than parcel B, and the convective scheme simply

switches their vertical positions (i.e., stacking order) to restore a stable

stratification. Adapted from Haertel and Straub (2010).


a small amount of tracer diffusion. However, only parcels

in the same density class are joined, and after the spinup is

complete the grouping of parcels is discontinued so that

there is absolutely no tracer diffusion away from the sur-

face. Moreover, the identities of individual parcels are

maintained during the grouping so that individual parcels

may be tracked through the merging and splitting process.

g. Sensitivity to model resolution

In order to test the sensitivity of our key results to model

resolution we carried out two supplemental simulations

that were initialized with the end state of the spinup

simulation and were continued after parcels were divided

either horizontally or vertically. These simulations were

run for 300 years to allow the upper ocean to adjust to the

new resolution. The upper-ocean density structure is sur-

prisingly robust, changing very little when the vertical

resolution is doubled or the horizontal resolution is dou-

bled (not shown). The gross structure of the meridional

overturning streamfunction is also the same for runs with

higher horizontal and vertical resolution (Fig. A3). The

deep cell is slightly more intense with a higher vertical

resolution (Fig. A3a), and the Deacon cell is stronger with

a higher horizontal resolution (Fig. A3b), presumably

because of a stronger meander in the ACC (Fig. 13d).

The LOM has been developed over a number of years,

and for more details on the numerical method, the mixing

parameterizations, and tests that compare LOM simula-

tions to analytic solutions and runs conducted with other

ocean models the reader is referred to Haertel and

Randall (2002) and Haertel et al. (2004, 2009). There are

only a few differences between the model used here and

that in Haertel et al. (2009): in this study, we use a Mer-

cator projection, a modified leapfrog time differencing

with implicit treatment of Coriolis terms, and the modi-

fied form of gravity wave retardation described above,

which has more straightforward energy conservation

properties.

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